A265742 Numbers k such that there exists at least an integer in the interval [e*k - 1/k, e*k + 1/k] where e = 2.71828... is Euler's number.
1, 2, 3, 4, 7, 14, 32, 39, 71, 142, 465, 536, 1001, 2002, 3003, 8544, 9545, 18089, 36178, 54267, 190435, 208524, 398959, 797918, 1196877, 4996032, 5394991, 10391023, 20782046, 31173069, 150869313, 161260336, 312129649, 624259298, 936388947, 1248518596, 5155334720
Offset: 1
Keywords
Examples
a(1) = 1 because there exist two integers (2 and 3) in the interval [1*e - 1/1, 1*e + 1/1] = [1.71828..., 3.71828...]; a(2) = 2 because the number 5 belongs to the interval [2*e - 1/2, 2*e + 1/2] = [4.93656..., 5.93656...]; a(3) = 3 because the number 8 belongs to the interval [3*e - 1/3, 3*e + 1/3] = [7.82151..., 8.48817...].
Links
- Takao Komatsu, The interval associated with a Fibonacci number, The Fibonacci Quarterly, Volume 41, Number 1, February 2003.
Programs
-
Maple
*** the program gives the interval [a,b],the integer(s) between [a,b] and k *** nn:=10^7: e:=exp(1): for n from 1 to nn do: x1:=evalhf(e*n-1/n):y1:=evalhf(e*n+1/n): x:=floor(x1):y:=floor(y1): for j from x+1 to y do: printf("%g %g %g %g\n",x1,y1,j,n): od: od:
Extensions
a(31)-a(32) corrected by and more terms from Jinyuan Wang, Jul 16 2025
Comments